On the Tu-Zeng Permutation Trinomial of Type $(1/4,3/4)$

Abstract

Let $q$ be a power of $2$. Recently, Tu and Zeng considered trinomials of the form $f(X)=X+a{X}^{(1/4)q^2(q-1)}+b{X}^{(3/4)q^2(q-1)}$, where $a,b\in {\mathbb{F}}_{q^2}^{\ast }$. They proved that $f$ is a permutation polynomial of ${\mathbb{F}}_{q^2}$ if $b=a^{2-q}$ and $X^3+X+a^{-1-q}$ has no root in ${\mathbb{F}}_q$. In this paper, we show that the above sufficient condition is also necessary.

arXiv comment24 pages

References in corpus (1)

• Determination of a Class of Permutation Trinomials in Characteristic Three

Cited by in corpus (2)

• On a conjecture about a class of permutation quadrinomials
• on a conjecture on permutation rational functions over finite fields